TITLE: MSO2-expressible graph properties unexpressible in MSO1
QUESTION [9 upvotes]: I would like to ask if there is any good list of graph properties that one can express in the MSO2 (i.e., using quantification over sets of edges) but provably not in MSO1 (i.e., using quantification over sets of vertices only). I know that this should be Hamilton path but I think that there has to be plenty such examples. Ideal are those that are suitable as exercises for non-expressibility results using Ehrenfeucht–Fraïssé games.

REPLY [3 votes]: I have found a book available online which gives some answers to your question. The book is Graph Structure and Monadic Second-Order Logic: a Language Theoretic Approach by Courcelle and Engelfriet. I am not (at all) an expert on this topic, but the book seems like a good reference. It lists the existence of a Hamiltonian cycle as an example. Some additional interesting examples included in the book are:

If a graph is simple
Existence of a perfect matching
Existence of a spanning tree with maximum degree 3

The most relevant parts of the book are Proposition 5.13, Proposition 5.19, and Remark 5.21. Showing the existence of a perfect matching is not expressible in $MSO_1$ can be with the same arguement on complete bipartite graphs $K_{n,m}$ outlined in Tony Huynh's answer. Also, it seems to me the example of a spanning tree with maximum degree 3 in Remark 5.21 can be modified to show that the existence of a spanning tree with maximum degree $k$ for some given $k$ in not expressible in $MSO_1$ (put more "leaves" on the graph in Figure 5.3).